Combining Positive and Negative Exponents

Date: 06/30/99 at 12:33:38
From: Lauren Fortner
Subject: Algebra: using the power theorem
I understand how to get problems from "beginning" form, for example:
x (x^-3)^2 y (xy^-2)^-3
-----------------------
(y^2)^3 y^-3 (x^2)^3
to "after" you use power theorem:
x x^-6 y x^-3 y^6
-----------------
y^6 y^-3 x^6
and then when you simplify the numerator and denominator:
x^-8 y^7
--------
y^3 x^6
What I don't understand is how you get from that to writing all
exponential expressions with positive exponents, or negative
exponents, or with both. I homeschool, so I don't really have anyone
to ask.
Thank you for your help.

Date: 07/01/99 at 17:00:53
From: Doctor Peterson
Subject: Re: Algebra: using the power theorem
Hi, Lauren. Thanks for a well-written question - it really helps to
know just what part you have trouble with.
The key to this step is that
x^-a 1 1 x^a
---- = --- and ---- = ---
1 x^a x^-a 1
That's essentially just the definition of negative exponents, and I'll
assume you're at least aware of this as a fact. What you need is how
to apply it.
What these facts mean in practice is that you can move a factor from
top to bottom or from bottom to top and negate the exponent. Once you
get used to it, you just think "There's an x^-8 in the numerator, so I
can replace that with an x^8 in the denominator." To take it more
slowly, we can pull the expression apart, apply the rule, and put it
back together:
x^-8 y^7 x^-8 y^7 1 1 1 y^7 1 1
-------- = ---- * --- * --- * --- = --- * --- * --- * ---
y^3 x^6 1 1 y^3 x^6 x^8 1 y^3 x^6
y^7
= -----------
x^8 y^3 x^6
That makes all the exponents positive, but there's still another step
you didn't mention: combining like factors so that x and y each appear
only once. That's easy in the denominator now; we can just permute so
the x's are together and add the exponents:
y^7 y^7
----------- = --------
x^8 y^3 x^6 x^14 y^3
But you still have y in two places. We can use the same rule to get
the y's together; since 7 > 3, let's move the y^3 to the top to keep
the exponents positive:
y^7 y^7 1 1 y^7 1 y^-3 y^7 y^-3 y^4
-------- = --- * ---- * --- = --- * ---- * ---- = -------- = ----
x^14 y^3 1 x^14 y^3 1 x^14 1 x^14 x^14
Now we're really done.
But we've taken a lot more steps than we had to. That's fine when
you're starting out; this isn't a race. But here's how I'd do it
myself:
x(x^-3)^2y(xy^-2)^-3 x x^-6 y x^-3 y^6
-------------------- = -----------------
(y^2)^3y^-3(x^2)^3 y^6 y^-3 x^6
= x x^-6 y x^-3 y^6 * y^-6 y^3 x^-6
(Here I've moved everything to the top.)
= x x^-6 x^-3 x^-6 * y y^6 y^-6 y^3
(Here I've gathered the x's and y's
together; I'd probably mark each factor
as I copied it to make sure I didn't miss
any.)
= x^(1-6-3-6) y^(1+6-6+3)
= x^-14 y^4
y^4
= ----
x^14
If you're at all afraid of negative exponents, this will make you
dizzy, but if you like to take the bull by the horns and get it over
with, making the negative exponents work for you, this is the way to
do it. Basically, I decided I'd rather deal with positive and negative
exponents than with numerators and denominators; they're really two
ways to say the same thing, and mixing them doesn't make sense. So
even though the goal is to have numerator and denominator and
eliminate negative exponents, I found that it's really easier to work
with the negatives, then change them to denominators when I'm done.
If any of this overwhelmed you, or if you have more questions, feel
free to write back.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/